Quantum logarithmic Sobolev inequalities and rapid mixing Michael - - PowerPoint PPT Presentation

Motivation Results Applications and outlook

Quantum logarithmic Sobolev inequalities and rapid mixing

Michael Kastoryano and Kristan Temme

1Dahlem Center for Complex Quantum Systems, Freie Universit¨

at Berlin, 14195 Berlin, Germany

2Center for Theoretical Physics, MIT, Cambridge, MA 02139, USA

QIP 2013 Beijing

January 21, 2013

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

Outline

1 Motivation

Setting Convergence rates

2 Results

Mixing times Mathematical results

3 Applications and outlook

Quantum Expanders Liouvillian Complexity

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Setting

Setting

• We consider only finite dimensional state spaces.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Setting

Setting

• We consider only finite dimensional state spaces.
• We consider an open quantum system described by a

Markovian master equation d dt ρt = L(ρ) = i[H, ρ] +

• k

LkρL†

k − 1

2{L†

kLk, ρ}

(1)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Setting

Setting

• We consider only finite dimensional state spaces.
• We consider an open quantum system described by a

Markovian master equation d dt ρt = L(ρ) = i[H, ρ] +

• k

LkρL†

k − 1

2{L†

kLk, ρ}

(1)

• We assume that the Liouvillian is primitive, meaning that L

has a unique full-rank stationary state σ > 0

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Setting

Setting

• We consider only finite dimensional state spaces.
• We consider an open quantum system described by a

Markovian master equation d dt ρt = L(ρ) = i[H, ρ] +

• k

LkρL†

k − 1

2{L†

kLk, ρ}

(1)

• We assume that the Liouvillian is primitive, meaning that L

has a unique full-rank stationary state σ > 0

• If ΓσL = L∗Γσ, where σ is the stationary state of L and

Γσ(X) = √σX√σ, the L is reversible.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Setting

Setting

• We consider only finite dimensional state spaces.
• We consider an open quantum system described by a

Markovian master equation d dt ρt = L(ρ) = i[H, ρ] +

• k

LkρL†

k − 1

2{L†

kLk, ρ}

(1)

• We assume that the Liouvillian is primitive, meaning that L

has a unique full-rank stationary state σ > 0

• If ΓσL = L∗Γσ, where σ is the stationary state of L and

Γσ(X) = √σX√σ, the L is reversible. Note: we do not yet make any assumptions about locality or geometry at this point.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

The question

Let L be the generator of a primitive reversible quantum dynamical

• semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have

||ρt − σ||1 ≤ ǫ? (2)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

The question

Let L be the generator of a primitive reversible quantum dynamical

• semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have

||ρt − σ||1 ≤ ǫ? (2)

The answer: general convergence theorem

Let λ > 0 be the spectral gap of L, then for any b ≤ λ, there exists a finite A such that ||ρt − σ||1 ≤ Ae−bt. (3)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

The question

Let L be the generator of a primitive reversible quantum dynamical

• semigroup. Given ǫ > 0, for what τ ≥ t > 0 do we have

||ρt − σ||1 ≤ ǫ? (2)

The answer: general convergence theorem

Let λ > 0 be the spectral gap of L, then for any b ≤ λ, there exists a finite A such that ||ρt − σ||1 ≤ Ae−bt. (3) What are good choices for A and b? We will argue that the Log Sobolev machinery is the finest available to answer this question.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

Applications

1 Unital quantum channels and random unitary maps (the fast

scrambling conjecture).

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

Applications

1 Unital quantum channels and random unitary maps (the fast

scrambling conjecture).

2 Quantum memories: Davies generators of stabilizer

• Hamiltonians. Rigorous no-go theorems.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

Applications

1 Unital quantum channels and random unitary maps (the fast

scrambling conjecture).

2 Quantum memories: Davies generators of stabilizer

• Hamiltonians. Rigorous no-go theorems.

3 Liouvillian complexity: what can we say about systems whose

Log Sobolev constant is independent of the system size?

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

Applications

1 Unital quantum channels and random unitary maps (the fast

scrambling conjecture).

2 Quantum memories: Davies generators of stabilizer

• Hamiltonians. Rigorous no-go theorems.

3 Liouvillian complexity: what can we say about systems whose

Log Sobolev constant is independent of the system size?

4 Dissipative algorithms?

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Convergence rates

Applications

1 Unital quantum channels and random unitary maps (the fast

scrambling conjecture).

2 Quantum memories: Davies generators of stabilizer

• Hamiltonians. Rigorous no-go theorems.

3 Liouvillian complexity: what can we say about systems whose

Log Sobolev constant is independent of the system size?

4 Dissipative algorithms? 5 Concentration of measure?

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

A few definitions to start with...

non-commutative Lp spaces

• The Lp inner product. For two hermitian operators f , g:

f , gσ = tr[Γσ(f )g] ≡ tr[σ1/2f σ1/2g]. (4)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

A few definitions to start with...

non-commutative Lp spaces

• The Lp inner product. For two hermitian operators f , g:

f , gσ = tr[Γσ(f )g] ≡ tr[σ1/2f σ1/2g]. (4)

• The Lp norm. For any hermitian operator f :

||f ||p,σ = tr[ |Γ1/p

σ

(f )|p]

1/p

(5)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

A few more definitions...

Variance and Entropy functionals

• The variance

Varσ(g) = tr[Γσ(g)g] − tr[Γσ(g)]2. (6)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

A few more definitions...

Variance and Entropy functionals

• The variance

Varσ(g) = tr[Γσ(g)g] − tr[Γσ(g)]2. (6)

• The Lp relative entropies. For any hermitian operator f :

Ent1(f) = tr[Γσ(f )(log(Γσ(f )) − log(σ))] (7) −tr[Γσ(f )] log(tr[Γσ(f )]) (8) Ent2(f) = tr[

• Γ1/2

σ (f )

2 log

• Γ1/2

σ (f )

• ]

(9) −1 2tr[

• Γ1/2

σ (f )

2 log (σ)] −1 2f 2

2,σ log

• f 2

2,σ

• .

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

Yet more... (sorry!)

Dirichlet Forms

E1(f ) = −1 2tr[Γσ(L(f ))(log(Γσ(f )) − log(σ))] (10) E2(f ) = − f , L(f )σ . (11) Useful identities: Var(Γ−1

σ (ρ)) = χ2(ρ, σ),

Ent2(Γ−1

σ (ρ)) = S(ρ||σ)

(12)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

Spectral Gap and Log-Sobolev constant

• The spectral gap of L:

λ = inf

f =0

E2(f ) Varσ(f) (13)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

Spectral Gap and Log-Sobolev constant

• The spectral gap of L:

λ = inf

f =0

E2(f ) Varσ(f) (13)

• The (1, 2)- logarithmic Sobolev constant

α1,2 = inf

f >0

E1,2(f ) Ent1,2(f) (14)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook

Spectral Gap and Log-Sobolev constant

• The spectral gap of L:

λ = inf

f =0

E2(f ) Varσ(f) (13)

• The (1, 2)- logarithmic Sobolev constant

α1,2 = inf

f >0

E1,2(f ) Ent1,2(f) (14) Note: one can in fact define a whole family of Log Sobolev constants αp, with p ≥ 0.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Mixing times

Theorem

Let L denote the generator of a primitive reversible semigroup with fixed point σ. Then,

1 χ2 bound:

||ρt − σ||1 ≤

• χ2(ρt, σ)

(15) ≤

• χ2(ρ, σ)e−λt ≤
• 1/σmine−λt.

Where σmin denotes the smallest eigenvalue of the fixed point σ.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Mixing times

Theorem

Let L denote the generator of a primitive reversible semigroup with fixed point σ. Then,

1 χ2 bound:

||ρt − σ||1 ≤

• χ2(ρt, σ)

(15) ≤

• χ2(ρ, σ)e−λt ≤
• 1/σmine−λt.

2 Log-Sobolev bound:

||ρt − σ||1 ≤

• 2S(ρt||σ)

(16) ≤

• 2S(ρ||σ)e−α1t ≤
• 2 log (1/σmin)e−α1t.

Where σmin denotes the smallest eigenvalue of the fixed point σ.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Mixing times

Interpretation

Davies generators describe the dissipative dynamics resulting as the weak (or singular) coupling limit of a system coupled to a large heat bath. For these thermal maps, the Log-Sobolev constant is the minimal normalized rate of change of the free energy of the system: α1 = inf

ρ ∂t log [F(ρt) − F(ρβ)]|t=0 ,

(17) where F(ρ) = tr[ρH] − 1

βS(ρ) is the free energy of the system, and

ρβ is the Gibbs state.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Mathematical results

Mathematical results

Theorem (Partial ordering)

Let L be a primitive reversible Liouvillian with stationary state σ. The Log-Sobolev constants α1, α2 and the spectral gap λ of L are related as: α2 ≤ α1 ≤ λ. (18)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Mathematical results

Theorem (Hypercontractivity)

Let L be a primitive Liouvillian with stationary state σ, and let Tt = etL be its associated semigroup. Then

1 If there exists a α > 0 such that ||Tt||(2,σ)→(p(t),σ) ≤ 1 for all

t > 0 and 2 ≤ p(t) ≤ 1 + e2αt. Then L satisfies LS2 with α2 ≥ α.

2 If L is weakly Lp-regular, and has an LS2 constant α2, then

||Tt||(2,σ)→(p(t),σ) ≤ 1 for all t > 0 when 2 ≤ p(t) ≤ 1 + e2α2t. If, furthermore, L is strongly Lp regular, then the above holds for all t > 0 when 2 ≤ p(t) ≤ 1 + e4α2t.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Quantum Expanders

Quantum expanders

Quantum Expander: (sequence of) quantum channel with i) a fixed number of Kraus operators (D), and ii) the spectral gap λ of the channel is asymptotically independent of dimension d. Then, (1 − 2/d)λ log (d − 1) ≤ α2 ≤ log D 4 + log log d 2 log 3d/4 (19)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Quantum Expanders

Quantum expanders

Quantum Expander: (sequence of) quantum channel with i) a fixed number of Kraus operators (D), and ii) the spectral gap λ of the channel is asymptotically independent of dimension d. Then, (1 − 2/d)λ log (d − 1) ≤ α2 ≤ log D 4 + log log d 2 log 3d/4 (19) The mixing time is of order log d

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Liouvillian Complexity

Suppose that L describes the open system dynamics on a lattice of

• qudits. Assume furthermore that L is: i) primitive and

reversible, ii) local, and iii) has a Log Sobolev constant α1 which is system size independent. Then we get

(strong) clustering of correlations

OAOBσ − OAσOBσ ≤ K log

• 1

σmin

• e−α1d(A,B)/v

(20) where K is volume like.

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Liouvillian Complexity

Suppose that L describes the open system dynamics on a lattice of

• qudits. Assume furthermore that L is: i) primitive and

reversible, ii) local, and iii) has a Log Sobolev constant α1 with is system size independent. Then we get

Stability of Liouvillians

Let Q be a local perturbation of L, and L′ = L + Q with stationary state σ′, then ||σ − σ′||1 ≤ ||Q||1−1 α1

• log
• log
• 1

σmin

• + 1
• (21)

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Liouvillian Complexity

Thank you for your attention!

Quantum LogSobolev Michael Kastoryano and Kristan Temme

Motivation Results Applications and outlook Liouvillian Complexity

References

MJK and Kristan Temme Quantum logarithmic Sobolev inequalities and rapid mixing. arXiv:1207.3261

• R. Olkiewicz, B. Zegarlinski

Hypercontractivity in noncommutative Lp spaces.

• J. Func. Analy. 161(1):246-285 (1999)
• K. Temme, MJK, M.B. Ruskai, M.M. Wolf, F. Verstraete

The χ2 divergence and mixing times of quantum Markov processes.

• J. Math. Phys. 51, 122201 (2010)

MJK, T. Osborne, J. Eisert, Correlations and Area laws for open quantum systems. upcoming

Quantum LogSobolev Michael Kastoryano and Kristan Temme